Exact location of the phase transition for random (1,2)-QSAT

Creignou, Nadia; Daudé, Hervé; Egly, Uwe; Rossignol, Raphaël

doi:10.1051/ita/2014025

Creignou, Nadia ¹ ; Daudé, Hervé ² ; Egly, Uwe ³ ; Rossignol, Raphaël ^4,⁵

¹ Aix-Marseille Université, CNRS, LIF UMR 7279, 13288 Marseille, France
² Aix-Marseille Université, CNRS, I2M UMR 7373, 13453 Marseille, France
³ Institut für Informationsysteme 184/3, Technische Universität Wien, Favoritenstrasse 9-11 A-1040 Wien, Austria
⁴ University Grenoble Alpes, IF, 38000 Grenoble, France
⁵ CNRS, IF, 38000 Grenoble, France

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 49 (2015) no. 1, pp. 23-45.

Résumé

The $Q S A T$ problem is the quantified version of the $S A T$ problem. We show the existence of a threshold effect for the phase transition associated with the satisfiability of random quantified boolean CNF formulas of the form $\forall X \exists Y ϕ (X, Y)$ , where $X$ has $m$ variables, $Y$ has $n$ variables and each clause in $ϕ$ has one literal from $X$ and two from $Y$ . For such formulas, we show that the threshold phenomenon is controlled by the ratio between the number of clauses and the number $n$ of existential variables. Then we give the exact location of the associated critical ratio $c^{*}$ : it is a decreasing function of $α$ , where $α$ is the limiting value of $m / l o g (n)$ when $n$ tends to infinity. Thus we give a precise location of the phase transition associated with a $c o N P$ -complete problem.

Reçu le : 2014-05-05
Accepté le : 2014-08-28

MR Zbl

DOI : 10.1051/ita/2014025

Classification : 68R01, 60C05, 05A16
Mots clés : Random quantified formulas, satisfiability, phase transition, sharp threshold

Affiliations des auteurs :

Creignou, Nadia ¹ ; Daudé, Hervé ² ; Egly, Uwe ³ ; Rossignol, Raphaël ^{4, 5}

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     title = {Exact location of the phase transition for random {(1,2)-QSAT}},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {23--45},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {1},
     year = {2015},
     doi = {10.1051/ita/2014025},
     mrnumber = {3342171},
     zbl = {1327.68129},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2014025/}
}

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Creignou, Nadia; Daudé, Hervé; Egly, Uwe; Rossignol, Raphaël. Exact location of the phase transition for random (1,2)-QSAT. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 49 (2015) no. 1, pp. 23-45. doi : 10.1051/ita/2014025. http://www.numdam.org/articles/10.1051/ita/2014025/

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